As such a device, a device which makes a driving voltage of a starter motor variable in accordance with a state of a vehicle is known. For example, in JP-2002-161838 A (corresponding to US 2002/0070555), in a case where the internal combustion engine is required to be restarted during the period of time when the internal combustion engine is automatically stopped by idling stop control, the driving voltage of a starter becomes higher than that in a case where the internal combustion engine is required to be started by a key operation of a driver.
However, as a technology for improving fuel efficiency, coasting control is known. The coasting control is a technology which stops combustion of an internal combustion engine by performing a fuel cut of the internal combustion engine, and allows a vehicle to perform inertial traveling by making a clutch device provided between the internal combustion engine and a transmission be in a power blocking state. In the middle of the coasting control, a traveling speed of the vehicle is relatively high. For this reason, when the internal combustion engine is required to be started in the middle of the coasting control, the rotation speed of the engine is required to be rapidly raised to a high rotation speed in accordance with the traveling speed of the vehicle after improving acceleration of the vehicle from the state of the coasting control without causing the driver to experience discomfort.